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Continuity And Differentiability

Question

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$space space space space space space space space space space space space space straight f left parenthesis straight x right parenthesis equals open parentheses fraction numerator 3 plus straight x over denominator 1 plus straight x end fraction close parentheses to the power of 2 plus 3 straight x end exponent therefore space space space log space straight f left parenthesis straight x right parenthesis equals log open parentheses fraction numerator 3 plus straight x over denominator 1 plus straight x end fraction close parentheses to the power of 2 plus 3 straight x end exponent rightwards double arrow space space log space straight f left parenthesis straight x right parenthesis equals left parenthesis 2 plus 3 straight x right parenthesis open parentheses fraction numerator 3 plus straight x over denominator 1 plus straight x end fraction close parentheses rightwards double arrow space space log space straight f left parenthesis straight x right parenthesis equals left parenthesis 2 plus 3 straight x right parenthesis left square bracket log left parenthesis 3 plus straight x right parenthesis minus log left parenthesis 1 plus straight x right parenthesis right square bracket Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma fraction numerator 1 over denominator straight f left parenthesis straight x right parenthesis end fraction. straight f apostrophe left parenthesis straight x right parenthesis equals left parenthesis 2 plus 3 straight x right parenthesis. open square brackets fraction numerator 1 over denominator 3 plus straight x end fraction minus fraction numerator 1 over denominator 1 plus straight x end fraction close square brackets plus 3. left square bracket log left parenthesis 3 plus straight x right parenthesis minus lg left parenthesis 1 plus straight x right parenthesis right square bracket Put space straight x equals 0 therefore space 1 over 9 straight f apostrophe left parenthesis 0 right parenthesis equals left parenthesis 2 plus 0 right parenthesis. open square brackets fraction numerator 1 over denominator 3 plus 0 end fraction minus fraction numerator 1 over denominator 1 plus 0 end fraction close square brackets plus 3. left square bracket log left parenthesis 3 plus 0 right parenthesis minus lg left parenthesis 1 plus 0 right parenthesis right square bracket space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space straight f left parenthesis 0 right parenthesis equals open parentheses fraction numerator 3 plus 0 over denominator 1 plus 0 end fraction close parentheses to the power of 2 plus 0 end exponent equals 3 squared equals 9 close square brackets rightwards double arrow space 1 over 9 straight f apostrophe left parenthesis 0 right parenthesis equals 2 open parentheses fraction numerator 1 minus 3 over denominator 3 end fraction close parentheses plus 3 left parenthesis log space 3 minus 0 right parenthesis rightwards double arrow space straight f apostrophe left parenthesis 0 right parenthesis equals negative 12 plus 27 space log space 3.$

Solution

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